scholarly journals An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

2019 ◽  
Vol 14 (2) ◽  
pp. 204
Author(s):  
Eduardo Hernandez-Montero ◽  
Andres Fraguela-Collar ◽  
Jacques Henry
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Laplace Equation ◽  
Boundary Data ◽  
A Priori ◽  
Normal Derivative ◽  
Cauchy Data ◽  
Dirichlet Data ◽  
Data Completion ◽  
The Cauchy Problem

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

The Cauchy problem for the system of Zakharov equations arising from ion-acoustic modes

1996 ◽  
Vol 126 (4) ◽  
pp. 811-820 ◽  
Author(s):  
Guo Boling ◽  
Yuan Guangwei
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Cauchy Data ◽  
Continuous Method ◽  
Initial Value ◽  
Acoustic Modes ◽  
Ion Acoustic ◽  
The Cauchy Problem

In this paper the initial value problem for a class of Zakharov equations arising from ion-acoustic modes is discussed. Without assuming the Cauchy data are small, we prove the existence and uniqueness of the global smooth solution for the problem via the so-called continuous method and delicate a priori estimates.

scholarly journals An Alternating Potential-Based Approach To The Cauchy Problem For The Laplace Equation In A Planar Domain With A Cut

2008 ◽  
Vol 8 (4) ◽  
pp. 315-335 ◽  
Author(s):  
R. CHAPKO ◽  
B.T. JOHANSSON
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Neumann Boundary ◽  
Normal Derivative ◽  
Mixed Problems ◽  
Simply Connected Domain ◽  
The Cauchy Problem ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Direct Problems

Abstract We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

scholarly journals An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

2011 ◽  
Vol 9 (4) ◽  
pp. 878-896 ◽  
Author(s):  
Houde Han ◽  
Leevan Ling ◽  
Tomoya Takeuchi
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Regularization Method ◽  
Boundary Data ◽  
Outer Boundary ◽  
Cauchy Problems ◽  
Electrical Field ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractDetecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

On the Cauchy Problem for the Three-Dimensional Laplace Equation

2021 ◽  
Vol 65 (2) ◽  
pp. 49-64
Author(s):  
A. B. Khasanov ◽  
F. R. Tursunov
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Three Dimensional ◽  
The Cauchy Problem
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